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As for activities with-in walking distance from our Harbor Apartment we suggest the following amongst other things:

The beautiful Catholic Cathedral at Landakot is just up the street from our location.
The Old Cathedral by the city square, Austurvöllur, is about 5 minutes away with the Parliament building right next to it.
Hallgrímskirkja church, is a wonderful Lutheran cathedral with an observation floor at the top of its tower and houses the largest organ in the country, a ca 15 minute walk.

Perlan (Pearl) sits on the cities hot water tanks and has an outdoor observation deck, 360 degrees (another 15 minute walk from Hallgrímskirkja, 30 minutes from apartment, but with bus services).

The new Harpa Concert and Conference Center, home of the Iceland Symphony Orchestra is 3 minutes away, a magnificent architecture. Here you find a list of concerts, shows and other activities https://en.harpa.is/events/

A visit to a Reykjavík swimming pool is a must with its geothermal hot water and sit-in pools with up to 42+ Celcius. Our neighborhood pool is the Vesturbæjarlaug near by, 10 minutes by foot and there is a pool in just about every neighbourhood in Reykjavík.
https://sundlaugar.is/?lang=en

Whale watching boats leave practically from our doorsteps at the Old Harbor!
https://elding.is

We reccomend a day-long bus-trip called the Golden Circle. In one day you will experience Thingvellir National Park, one of the oldest parliament sites in the world, the original Geysir and its younger brother Strokkur and see how they compare to Old Faithful! and one of the most stunning waterfalls in Iceland, the Golden Waterfall!
https://www.tour.is/news/what-is-the-golden-circle-in-iceland

Some guest opt to visit the Blue Lagoon upon arrival, since its close to the International Airport, KEF. There is bus service from the airport to the Blue Lagoon and then from the lagoon to the city. Others have chosen to check-out early and do this on their way to the airport. An unforgettable experience!
https://www.bluelagoon.com

For great dining you have the Steikhúsið steakhouse just around the corner from Harbor Apartment, just turn right on Geirsgata street and in a minute you are there! http://steik.is/vefur/

Just a little further on the same street are two terrific fish and chips restaurants for lighter fare.

Owned by same award winning chef Hrefna Sætran are the Grillmarkaðurinn grill market and the Fiskmarkaðurinn fish market, both very near.
https://www.grillmarkadurinn.is
http://www.fiskmarkadurinn.is